Finite Group Databasebeta

← #11781812

Subgroups — #11781812

Order 1152.

351 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10592723C211yesmaximal, characteristic
2C3×#5020C212yesmaximal
3#10590387C211yesmaximal, characteristic
4#10589098C211yesmaximal, characteristic
5#10589092C212yesmaximal
6#71121—32nomaximal
7C32×C4×Q8C2211yescharacteristic
8C3×#869C2212yes—
9C2×#3718—21no—
10C2×#3718—21no—
11C4⋊(C3⋊(C3⋊C8))—21no—
12C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13#63630—21no—
14C4×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#835—22no—
16C3×#808—22no—
17C3×C4⋊(C3⋊C8)C2212yes—
18C3×#475—32no—
19#5092—32no—
20#5024—32no—
21#5024—61no—
22#5020D312yes—
23#4547—32no—
24#4486—32no—
25#4480—32no—
26C2×C32×Q8D411yescharacteristic
27C2×C3×Dic6—22no—
28C32×C4⋊C4—21no—
29C32×C4⋊C4D411yescharacteristic
30C32×C42—21no—
31C32×C42C2311yescharacteristic
32C2×C3⋊(C3⋊C8)—21no—
33C2×C3⋊(C3⋊C8)D411yescharacteristic
34C2×C3⋊(C3⋊C8)D411yescharacteristic
35C3×C4⋊Dic3—22no—
36C3×C4⋊Dic3—22no—
37C2×C3×C3⋊C8—22no—
38#3718—42no—
39#3718—42no—
40#1690—91noSylow
41C3×C4⋊Q8—32no—
42C3×C4×Q8—21no—
43C3×C4×Q8C2×D312yes—
44C2×#216—62no—
45C2×#216—62no—
46#869C2×D312yes—
47C3×#106—62no—
48C3×C4⋊C8—32no—
49C3×#103—62no—
50#835—22no—
51#808—22no—
52C4⋊(C3⋊C8)—32no—
53C4⋊(C3⋊C8)—61no—
54C4⋊(C3⋊C8)—62no—
55C4⋊(C3⋊C8)—62no—
56C4⋊(C3⋊C8)C2×D312yes—
57#803—62no—
58#803—62no—
59C4×C3⋊C8—32no—
60C4×C3⋊C8—61no—
61C32×Q8—22no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4C2×D411yescharacteristic
65C2×C32×C4C2×D411yescharacteristic
66C2×C32×C4#4111yescharacteristic
67C2×C3×Dic3—42no—
68C3×Dic6—42no—
69C3×Dic6—42no—
70C3⋊(C3⋊C8)—21no—
71C3⋊(C3⋊C8)—41no—
72C3⋊(C3⋊C8)SD1612yes—
73C3×C3⋊C8—42no—
74#494—91no—
75#475—32no—
76#446—91no—
77#334—91no—
78#332—92no—
79C2×C3×Q8—21no—
80C2×C3×Q8—62no—
81C2×C3×Q8#6712yes—
82C2×Dic6—22no—
83C2×C3×C8—62no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—21no—
87C3×C4⋊C4—62no—
88C3×C4⋊C4—62no—
89C3×C4⋊C4#6712yes—
90C3×C42—22no—
91C3×C42—22no—
92C3×C42—21no—
93C3×C42C22×D312yes—
94#216—124no—
95#216—124no—
96C4⋊Dic3—22no—
97C4⋊Dic3—22no—
98C2×C3⋊C8—22no—
99C2×C3⋊C8—32no—
100C2×C3⋊C8—32no—
101C2×C3⋊C8—61no—
102C2×C3⋊C8—61no—
103C2×C3⋊C8—62no—
104C2×C3⋊C8—62no—
105C22×C32#12111yescharacteristic
106C32×C4—21no—
107C32×C4—41no—
108C32×C4—41no—
109C32×C4C2×SD1611yescharacteristic
110C32×C4AGL(1,ℤ/8)11yescharacteristic
111C32×C4#13712yes—
112C3×Dic3—82no—
113C2×Q16—181no—
114C2×Q16—181no—
115C4⋊Q8—32no—
116C4×Q8D3211yescharacteristic
117#106—62no—
118C4⋊C8—32no—
119C4⋊C8—91no—
120C4⋊C8—181no—
121#103—62no—
122#97—181no—
123C4×C8—91no—
124C3×Q8—22no—
125C3×Q8—22no—
126C3×Q8—42no—
127C3×Q8—122no—
128C3×Q8—122no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—122no—
137C2×C3×C4D3×D412yes—
138C2×C3×C4#23712yes—
139C2×C3×C4C2×#6712yes—
140C2×Dic3—42no—
141Dic6—42no—
142Dic6—42no—
143C3×C8—122no—
144C3⋊C8—34no—
145C3⋊C8—42no—
146C3⋊C8—62no—
147C3⋊C8—62no—
148C3⋊C8—121no—
149C3⋊C8—122no—
150C3⋊C8—122no—
151C2×C32#46011yescharacteristic
152C2×C32#47011yescharacteristic
153C2×C32#47311yescharacteristic
154C2×Q8—62no—
155C2×Q8#62511yescharacteristic
156Q16—362no—
157Q16—362no—
158C2×C8—62no—
159C2×C8—91no—
160C2×C8—91no—
161C2×C8—181no—
162C4⋊C4—21no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4#62511yescharacteristic
166C42—21no—
167C42C2×D3211yescharacteristic
168C22×C3—21no—
169C22×C3#93912yes—
170C3×C4—22no—
171C3×C4—22no—
172C3×C4—21no—
173C3×C4—21no—
174C3×C4—41no—
175C3×C4—42no—
176C3×C4—42no—
177C3×C4—42no—
178C3×C4—42no—
179C3×C4—242no—
180C3×C4D3×SD1612yes—
181C3×C4#91412yes—
182C3×C4#94212yes—
183C3×C4#95112yes—
184Dic3—82no—
185C32#169011yescharacteristic, complemented, Sylow
186Q8—22no—
187Q8—122no—
188Q8—122no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—122no—
192C2×C4#380511yescharacteristic
193C2×C4C2×#62511yescharacteristic
194C2×C4#381111yescharacteristic
195C8—92no—
196C8—122no—
197C8—181no—
198C8—361no—
199C2×C3—21no—
200C2×C3—21no—
201C2×C3—21no—
202C2×C3#505212yes—
203C2×C3#516312yes—
204C2×C3#520412yes—
205C22#6398011yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—242no—
210C4#6382712yes—
211C4#6394011yescharacteristic
212C4#6394111yescharacteristic
213C3—21no—
214C3#7112112yescomplemented
215C2#1059118911yescharacteristic
216C2#1059162211yescharacteristic
217C2#1059168311yescharacteristic
subgroupG/H
centerC22#63980
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socle, prime radicalC22×C32#121
hypercenterC4×Q8D32
hypocenterC32#1690
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
185. C32#1690semidirect
214. C3#71121semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#105927232
C3×#50202
#105903872
#105890982
#105890922
#711213